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For a normed space (X,||·||), define X^~ to be the set of all equivalent classes of Cauchy sequences obtained by the relation {x_n}∼{y_n} if and only if lim_(n)||x_n-y_n||=0. ...

Given a complex Hilbert space H with associated space L(H) of continuous linear operators from H to itself, the bicommutant M^('') of an arbitrary subset M subset= L(H) is ...

Binet's first formula for the log gamma function lnGamma(z), where Gamma(z) is a gamma function, is given by for R[z]>0 (Erdélyi et al. 1981, p. 21; Whittaker and Watson ...

The term Borel hierarchy is used to describe a collection of subsets of R defined inductively as follows: Level one consists of all open and closed subsets of R, and upon ...

A C^infty function is a function that is differentiable for all degrees of differentiation. For instance, f(x)=e^(2x) (left figure above) is C^infty because its nth ...

A Cartesian tensor is a tensor in three-dimensional Euclidean space. Unlike general tensors, there is no distinction between covariant and contravariant indices for Cartesian ...

Given a complex Hilbert space H with associated space L(H) of continuous linear operators from H to itself, the commutant M^' of an arbitrary subset M subset= L(H) is the ...

A Banach algebra A is called contractible if H^1(A,X)=Z^1(A,X)/B^1(A,X)=0 for all Banach A-bimodules X (Helemskii 1989, 1997). A C^*-algebra is contractible if and only if it ...

A derivation is a sequence of steps, logical or computational, from one result to another. The word derivation comes from the word "derive." "Derivation" can also refer to a ...

Dirichlet's principle, also known as Thomson's principle, states that there exists a function u that minimizes the functional D[u]=int_Omega|del u|^2dV (called the Dirichlet ...